Perfect Mendelsohn designs with equal-sized holes and block size four

✍ Scribed by F. E. Bennett; Zhang Xuebin

• Publisher: John Wiley and Sons
• Year: 1997
• Tongue: English
• Weight: 149 KB
• Volume: 5
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/(sici)1520-6610(1997)5:3<203::aid-jcd4>3.0.co;2-j

✦ Synopsis

Let M = {m1 , m2 , . . . , m h } and X be a v-set (of points). A holey perfect Mendelsohn designs (briefly (v, k, λ) -HPMD), is a triple (X, H, B), where H is a collection of subsets of X (called holes) with sizes M and which partition X, and B is a collection of cyclic k-tuples of X (called blocks) such that no block meets a hole in more than one point and every ordered pair of points not contained in a hole appears t-apart in exactly λ blocks, for 1 ≤ t ≤ k -1. The vector (m1 , m2 , . . . , m h ) is called the type of the HPMD.

we write briefly m h for the type. In this article, it is shown that the necessary condition for the existence of a (v, 4, λ) -HPMD of type m h , namely, λh(h -1)m 2 ≡ 0 (mod 4) is also sufficient with the exception of types 2 4 and 1 8 with λ = 1, and type m 4 for odd m with odd λ.